(2) It is clear that \(R=eR\oplus (1-e)R\)  and \(M=eM\oplus (1-e)M\) . By [1]}, \(A=R\propto M\cong (eR\propto eM)\times ((1-e)R\propto (1-e)M)\) 
(as rings). But \(eM=0\) , so the ring \(A\)  is isomorphic to the ring \(eR\times ((1-e)R\propto (1-e)M)\) . The first assertion is obvious.
\(\Box \)
